Wireless receiver

ABSTRACT

The present invention relates to a method and apparatus for demodulation in a wireless communications system transmitted across a wireless communications channel. The described wireless receiver includes a first antenna for receiving a wireless signal including a symbol transmitted across a wireless communications channel perceived by the first antenna, an observation modifier for generating a modified observation (y) of the symbol based on a product of the received observation (r) and the complex conjugate of a channel estimate (h*), a log-likelihood ratio (LLR) module generating log-likelihood ratios (LLRs) based on the modified observation and the channel estimate, and a maximum-likelihood-based decoder for decoding the symbol based on the LLRs.

CLAIM OF PRIORITY

This application claims the benefit of priority of Australia PatentApplication No. 2015904911, filed on Nov. 27, 2015, the benefit ofpriority of which is claimed hereby, and which is incorporated byreference herein in its entirety.

FIELD OF THE INVENTION

The present invention relates to a method and apparatus for demodulationin a wireless communications system. More particularly, the presentinvention relates to a method and apparatus for demodulating modifiedobservations of a symbol transmitted across a wireless communicationschannel.

BACKGROUND OF THE INVENTION

Referring to FIG. 1A, in a wireless communications system, a transmitter110 and a receiver 120 communicate via a wireless communications channel150. The transmitter may include an encoder 112 which encodes inputinformation bits into coded bits, and a modulator 118 which modulatesthe coded bits into a suitable signal format at radio-frequency (RF)frequencies for wireless transmission by the transmitting antennathrough the communications channel 150.

In practice, the effects of the communication channel 150 are to distortthe RF signal by, for example, introducing multipath effects, noise,timing jitters and frequency offsets. The role of the encoder 112 is toadd redundancy to the transmitted data so that errors due to such signaldistortions can be corrected after the distorted RF signal is receivedand demodulated at the receiver 120.

At the receiver 120, a receiving antenna receives the distorted RFsignal. The receiver 120 includes a demodulator 122 to demodulate thereceived RF signal to generate received bits 124. The received bits 124generally differ from the coded bits 116 due to the signal distortions.The receiver 120 also includes a decoder 126 to decode the receivedbits. The decoding process generates an estimate 128 of the informationbits 114 by essentially reversing the operation of decoder and, in doingso, attempting to recover the information bits 114 in the presence ofthe signal distortions. The receiver 120 further includes a channelestimator 130 to counteract the effects of the channel 150. FIG. 1Aillustrates that the decoder 126 in the illustrated example includes aViterbi decoder preceded by a block for preparing inputs for thecomputation of branch metrics. The decoder 126 estimates the informationbearing bits, which are re-encoded to allow use as training symbols inchannel estimation. The accuracy of the estimated channel will determinethe performance of the receiver 120. The transmitter/receiver mayinclude other components which are omitted from FIG. 1A, such as aninterleaver/de-interleaver, a scrambler/de-scrambler, and apuncturer/depuncturer.

In the case of an orthogonal frequency division multiplxing (OFDM) basedsystem, the received jth subcarrier in the ith data bearing OFDM symbolof a packet is modelled as

r[i,j]=h[i,j]d[i,j]+n[i,j]  (1)

where h[i,j] is a complex number representing the frequency domainchannel affecting subcarrier j in symbol i, d[i,j] is the symbol sent atthe transmitter, and n[i,j] is additive white Gaussian noise (AWGN)affecting the subcarrier. That is, r represents an observation of areceived symbol d which has been subject to rotation and scaling (viamultiplication by complex-valued h) and additive noise n.

In one implementation, the demodulation includes channel equalisationwhere the received subcarrier r[i,j] has the channel effectscounteracted (or zero-forced) via division as follows:

y _(equalised) [i,j]=r[i,j]/h[i,j]  (2).

The equalised or zero-forced observation y_(equalised)[i,j] is thenprovided to the decoder for decoding.

Reference to any prior art in the specification is not, and should notbe taken as, an acknowledgment or any form of suggestion that this priorart forms part of the common general knowledge in any jurisdiction orthat this prior art could reasonably be expected to be understood,regarded as relevant and/or combined with other pieces of prior art by aperson skilled in the art.

SUMMARY OF THE INVENTION

According to a first aspect of the invention, there is provided a methodcomprising: receiving, at a first antenna, an observation of a symboltransmitted across a wireless communications channel perceived by thefirst antenna; generating a modified observation based on a product ofthe received observation and the complex conjugate of a channel estimateof the channel; and generating, based on the modified observation andthe channel estimate, log-likelihood ratios (LLRs) for amaximum-likelihood-based decoder to decode.

Generating log-likelihood ratios may include generating a LLR associatedwith a most significant bit of the symbol. The log-likelihood ratio(LLR) associated with the most significant bit may be generated based onthe ratio of the real part (y_(I)) or imaginary part (y_(Q)) of themodified observation to Gaussian-distributed noise power of the channel.The LLR associated with the most significant bit may be estimated to be2 y₁/σ² or 2 y_(Q)/σ², where σ² represents the Gaussian-distributednoise power of the channel. Alternatively the LLR associated with themost significant bit may be estimated to be 4 y_(I)/σ² or 4 y_(Q)/σ².Still alternatively the LLR associated with the most significant bit maybe estimated to be 4 y_(I)/σ²−4|h|²/σ² or 4 y_(Q)/σ²−4|h|²/σ², where |h|represents the magnitude of the channel estimate. Still alternativelythe LLR associated with the most significant bit may be 4y_(I)/σ²+4|h|²/α².

Generating log-likelihood ratios may include generating a LLR associateda next most significant bit of the symbol. The LLR associated with thenext most significant bit may be generated based on the ratio of thereal part (y_(I)) or imaginary part (y_(Q)) of the modified observationto Gaussian-distributed noise power of the channel. The LLR associatedwith the most significant bit is 4|h|/σ²−2 y_(I)/σ² or 4|h|/σ²−2y_(Q)/σ².

The Gaussian-distributed noise power may be estimated or measured.

The method may further comprise receiving, at a second antenna locatedseparately from the first antenna, another observation of the symboltransmitted across a wireless communications channel perceived by thesecond antenna, and wherein generating a modified observation includesgenerating the modified observation also based on a product of the otherreceived observation and the complex conjugate of a channel estimate ofthe channel perceived by the second antenna. Generating the modifiedobservation may be based on a weighted sum of (a) the product of thereceived observation and the complex conjugate of a channel estimate ofthe channel and (b) the product of the other received observation andthe complex conjugate of a channel estimate of the channel perceived bythe second antenna. The weighted sum may be based on weights associatedwith the noise power as observed by the respective antennas.

The method may further comprise decoding, by a maximum-likelihood-baseddecoder, the LLRs.

According to a second aspect of the invention, there is provided awireless receiver comprising: a first antenna for receiving anobservation including a symbol transmitted across a wirelesscommunications channel perceived by the first antenna; an observationmodifier for generating a modified observation based on a product of thereceived observation and the complex conjugate of a channel estimate ofthe channel; a log-likelihood ratio (LLR) module generatinglog-likelihood ratios (LLRs) based on the modified observation; and amaximum-likelihood-based decoder for decoding the symbol based on theLLRs.

The wireless receiver may further comprise a second antenna forreceiving another observation of the symbol transmitted across awireless communications channel by the second antenna, the first antennaand the second antenna are located to provide spatial diversity, whereinthe observation modifier is configured to generate the modifiedobservation based on weighted sum of (a) the product of the receivedobservation and the complex conjugate of a channel estimate of thechannel and (b) the product of the other received observation and thecomplex conjugate of a channel estimate of the channel perceived by thesecond antenna.

The weighted sum may be based on weights associated with the noise poweras observed by the respective antennas.

The wireless receiver may further comprise a maximum-likelihood-baseddecoder for decoding the LLR.

According to a third aspect of the invention, there is provided anon-transitory machine-readable medium comprising instructionsexecutable by one or more processors, the instructions including thesteps according to the method of the first aspect.

Further aspects of the present invention and further embodiments of theaspects described in the preceding paragraphs will become apparent fromthe following description, given by way of example and with reference tothe accompanying drawings.

BRIEF DESCRIPTION OF THE DRAWINGS

FIG. 1 illustrates a schematic diagram of a wireless communicationssystem including a transmitter and a receiver.

FIGS. 2A and 2B illustrate simulated 16-QAM constellation diagrams of(A) equalised received symbols and (B) LLR-processed(log-likelihood-ratio-processed) received symbols under highchannel-power-to-noise-power, or signal-to-noise, ratio (SNR).

FIGS. 3A and 3B illustrate simulated 16-QAM constellation diagrams of(A) equalised received symbols and (B) LLR-processed received symbolsunder medium channel-power-to-noise-power, or signal-to-noise, ratio(SNR).

FIGS. 4A and 4B illustrate simulated 16-QAM constellation diagrams of(A) equalised received symbols and (B) LLR-processed received symbolsunder low channel-power-to-noise-power, or signal-to-noise, ratio (SNR).

FIG. 5 illustrates bit to constellation mapping of 16-QAM system.

FIG. 6A illustrates a wireless transmitter and of a wireless receivercoupled by a communication channel in accordance with one embodiment.

FIG. 6B illustrates a logic block diagram of an example of a subsystem600 in accordance with one embodiment.

FIG. 6C illustrates a method of demodulation in accordance with oneembodiment.

FIG. 6D illustrates a wireless transmitter and of a wireless receivercoupled by a communication channel in accordance with anotherembodiment.

FIG. 7A plots a comparison between optimal LLR values and approximatedLLR values for b0 for “small-magnitude” y_(I).

FIG. 7B plots a comparison between optimal LLR values and approximatedLLR values for b0 for “large-magnitude” y_(I).

FIG. 7C plots a comparison between optimal LLR values and approximatedLLR values for b1.

FIG. 7D plots a comparison between optimal LLR values and folding LLRvalues for b0.

FIG. 7E plots the comparison shown in FIGS. 7A, 7B, 7C and 7D together.

FIG. 8 plots a comparison of the packet error rate between usingsmall-magnitude approximation only and using both small andlarge-magnitude approximations to LLR.

FIG. 9 illustrates a logic block diagram of an example of a demodulatorin accordance with one embodiment.

DETAILED DESCRIPTION OF EMBODIMENTS Equalisation in Varying Channels

The inventors have recognised that the equalisation characterised byEquation (2) may give rise to degradation in decoding performance if thechannel is time-varying due to, for example, changing multipath effectsarising from movements of the receiver, the transmitter and/or anysurrounding objects.

Referring to Equation (1), the symbol d may be formed at the transmitterbased on a sequence of coded bits. For example, 1, 2, 4 or 6 coded bitsmay map to binary phase-shift keying (BPSK), quadrature phase-shiftkeying (QPSK), 16-QAM (quadrature amplitude modulation) or 64-QAMconstellations, respectively. Noise n is characterised by its power, andmay be estimated or measured at the receiver. For receivers implementingmaximum-likelihood based decoding, such as Viterbi decoding, theinventors recognise that degradation in decoding performance can ariseif the noise statistics of the inputs to the Viterbi decoder are notconstant over the received coded bits. The degradation can be attributedto the reliance of maximum-likelihood based estimation of the bitsequence on the product of Gaussian-distributed noise statistics. Anillustrative example using BPSK is described as follows.

In the case of BPSK, the Gaussian-distributed probability of theadditive white Gaussian noise (AWGN) for each of the two possiblesymbols for a respective subcarrier j can be characterised by:

$\begin{matrix}{C\; {\exp\left\lbrack \frac{{{r_{i} - {h_{i}d_{i}}}}^{2}}{2\sigma^{2}} \right\rbrack}} & (3)\end{matrix}$

over all bits constituting the bit sequence, where i now indexes the bitposition in the sequence, C is a constant, σ² represents the noise powerand |r_(i)−h_(i)d_(i)| represents the noise-induced deviation in the I-Q(in-phase and quadrature) plane between the received bit (r_(i)) and thetransmitted bit affected by a noise-free channel (h_(i)d_(i)). Forhigher order constellations, a similar sequence metric can be formed.

In practice, Viterbi decoding is executed in the logarithmic domain.Instead of computing a product of exponentials, therefore, a sum ofexponents is computed to provide the Viterbi metric in the followingform:

$\begin{matrix}{\sum_{i}\left\lbrack \frac{{{r_{i} - {h_{i}d_{i}}}}^{2}}{2\sigma^{2}} \right\rbrack} & (4)\end{matrix}$

where, for the purposes of calculating the metric, C is ignore since itis a constant for all sequence {d_(i)} considered. Viterbi decodinginvolves, among other steps, considering all possible bit sequences andselecting the one with the highest likelihood (or least distance in thelogarithmic domain). If σ is constant, it too can be disregarded as itwill not affect the selection based on sequence metrics. In practice,the Viterbi decoding may therefore be reduced to accumulators of theinputs, such that they can be run faster to inspect the possible bitsequences, albeit in a highly structured manner.

If a received observation is equalised, the equalised observation yaccording to Equation (2) is given by

y _(equalised) =r/h=d+n/h  (5)

In the case where the channel h is varying in amplitude (e.g. in thecase of fading channels), the effective noise power presented at theinput of the Viterbi decoder (i.e. σ² scaled based on h) is no longeridentically distributed from bit to bit and the Viterbi metric cannot besimplified as above without loss of performance. To illustrate such aloss in performance, FIGS. 2A-4B illustrate simulated 16-QAMconstellation diagrams of (a) received symbols equalised according toEquation (2) (labelled “Equalised”) and (b) received symbols processedas log-likelihood ratios (LLR) in preparation for maximum-likelihoodbased decoding (labelled “Maximum-likelihood” and using asmall-amplitude LLR approximation for the most significant bitsdescribed later) under high, medium, and lowchannel-power-to-noise-power, or signal-to-noise, ratio (SNR) at thereceiver. Each dot plotted in the I-Q plane represents one of multiplereceived symbols constructed from a random sequence of input bitstransmitted over a fading channel. The scattered pattern at each of the16 constellation points (see, for example, FIG. 2A) arises from theAWGN. The vertical and horizontal scales of these plots are arbitrary.The received symbols are assumed to be transmitted via a fading channel.As illustrated in FIGS. 2A and 2B, under high SNR (around 20 dB), both(a) and (b) have a recongisable 16-QAM constellation structure. Undermedium SNR (around 10 dB), only (a) has a recongisable 16-QAMconstellation structure, as seen in FIGS. 3A and 3B. Under low SNR(around 5 dB), neither (a) nor (b) has a recongisable 16-QAMconstellation structure, as seen in FIGS. 4A and 4B.

Having recognised the effects of varying effective noise from bit to biton maximum-likelihood based decoding performance, the inventors havedevised a method of demodulation based on a channel-matched observationto address these effects. The present disclosure is generally applicableto any QAM schemes, although each QAM scheme may be more efficientlyimplementation due to specific bit-to-constellation mapping. Thefollowing description discloses bit-to-constellation mapping using16-QAM as an example. Similar bit-to-constellation mapping may bederived for higher-order QAM schemes.

Bit to Constellation Mapping

In 16-QAM, constellation symbols d are defined by four bits in theordered sequence of b0, b1, b2 and b3, each representing a logical valueof either 0 or 1, as shown in FIG. 5, and taking a real value of either−1 or 1, respectively, as used in Equations (6a)-(8b) below.

The real part d_(I) (or the I-component) and the imaginary part d_(Q)(or Q-component) of the symbol d may be given by:

d _(I)=2b0−b0b1=b0(2−b1)  (6a)

d _(Q)=2b2−b2b3=b2(2−b3)  (6b)

The absolute values of each component are given by:

|d _(I) |=|b0(2−b1)|=|2−b1|=2−b1  (7a)

|d _(Q) |=|b2(2−b3)|=|2−b3|=2−b3  (7b)

Re-arranging (7a) and (7b) provides:

b1=2−|d _(I)|  (8a)

b3=2−|d _(Q)|  (8b)

Equations (8a) and (8b) therefore illustrate that it is possible torecover the least significant bits b1 and b3 from the absolute value ofthe I and Q components.

In 16-QAM, b0 and b2 are referred to as the most significant bits,whereas b1 and b3 are referred to as the next (or the least) significantbits. In 64-QAM (where a symbol constitutes 6 bits in the orderedsequence of b0, b1, b2, b3, b4 and b5), b0 and b3 are the mostsignificant bits, b1 and b4 are the next significant bits and b2 and b5are the least significant bits.

A Wireless Receiver and a Demodulator

Disclosed herein is a wireless receiver 220 as illustrated in FIG. 6A.The described wireless receiver 220 includes a first antenna 221 forreceiving a wireless signal including a symbol transmitted across awireless communications channel 250 perceived by the first antenna, anobservation modifier 222 for generating a modified observation (v) ofthe symbol based on a product of the received observation (r) and thecomplex conjugate of a channel estimate (h*), a log-likelihood ratio(LLR) module 225 generating log-likelihood ratios (LLRs) based on themodified observation and the channel estimate, and amaximum-likelihood-based decoder 226 for decoding the symbol based onthe LLRs. Together, the observation modifier 222 and the LLR module 225may be referred to as a demodulator 223. The receiver 220 may beconfigured to perform a demodulation method 300 as illustrated in FIG.6C. The outputs of the method 300 are log-likelihood ratios (LLRs) basedon modified observations of a symbol for subsequent decoding by amaximum-likelihood-based decoder, such as 226. The described method 300includes the step 302 of receiving, at a first antenna 221, anobservation of a symbol transmitted across a wireless communicationschannel 250, the step 304 of generating a modified observation based ona product of the received observation (r) and the complex conjugate of achannel estimate (h*), and the step 306 of generating, based on themodified observation, log-likelihood ratios (LLRs) for amaximum-likelihood-based decoder, such as the decoder 226, to decode.Steps 304 and 306 may be carried out by the demodulator 223.

The wireless communications channel 250 is one that is perceived by thefirst antenna 221. In some arrangement, as illustrated in FIG. 6D,wireless receiver 220 includes a second antenna 221′ located in relationto the first antenna 221 to provide spatial diversity. The secondantenna 221′ when so placed perceives a wireless communications channel250′ that is different from the channel 250 perceived by the firstantenna and is associated with a different channel estimate 230′. Ingeneral, the first antenna 221 and the second antenna 221′ are locatedat least approximately a wavelength of the wireless signal apart forspatial diversity.

In one arrangement, the wireless receiver 220 further includes, for eachantenna, a channel estimator 230 for providing the channel estimate anda noise power estimator 240 for estimating or measuring the noise powerof the perceived channel. As described further below, the observationmodifier 222 uses the channel estimate to generate the modifiedobservation, whereas the LLR module 225 uses the channel estimate andthe estimated or measured noise power to generate LLRs. A wirelessreceiver focussing on a single-antenna implementation is firstdescribed. A wireless receiver generalising to a multiple-antenna (i.e.spatially diversified) implementation is then described.

Channel-Matching an Observation

In the step 304, the observation modifier 222 is configured to generatea modified observation y. The observation modifier 222 computes themodified observation y based on a product of the received observation rand the complex conjugate of the complex-valued channel estimate h*. Inone implementation, the modified observation y is given by:

y=rh*  (9)

In another implementation, y is given by=crh*, that is, multiplied by anadditional constant c compared to the implementation given in Equation(9). In the description that follows, the modified observation y isassume to be given by Equation (9).

As will become apparent below, the modified observation y is“channel-matched” since the log-likelihood ratios calculated based onthe modified observation y have reduced dependence on the amplitude ofthe complex-valued channel h, compared to the equalised observationgiven by Equation (2). Stronger channels should generally be moreassertive than weaker channels. The observation is brought into coherentalignment with the original transmitted symbol, where the rotationaleffects because of the channels are removed.

Generating Log Likelihood Ratios

In step 306, the modified (e.g. channel-matched) observation generatedmay be provided to the LLR module 225 for generating LLRs. The generatedLLRs may be used for subsequent decoding by, for example, decoder 226.In one arrangement, the LLR module 225 computes the LLR for individualbits constituting the symbol d based on the modified observation y.Using the case of 16-QAM as an example, the probability P that thereceived observation r resulted from transmission with the i-th bitb_(i) (where iε0, 1, 2 and 3) defining the transmitted symbol d andbeing 1 is:

P(r|b _(i)=1)=Σ_(dεD) _(i) ₁ P(r|d)  (10)

where D_(i) ¹ is the set of all constellation points with bit i (b_(i))equal to 1. Expanding and simplifying (10) provides:

$\begin{matrix}{{P\left( {{rb_{i}} = 1} \right)} = {C\; {\sum\limits_{d \in D_{i}^{1}}{\exp\left\lbrack \frac{{{Re}\left( {y^{*}d} \right)} - \frac{{h}^{2}{d}^{2}}{2}}{\sigma^{2}} \right\rbrack}}}} & (11)\end{matrix}$

where y=rh* is the matching of the observation to the channel. A similarexpression to Equation (11) can be derived where the additional constantc is involved.

Decomposing the symbol constellation point d and the modifiedobservation y into their in-phase (I) and quadrature (Q) components asd=d_(I)+jd_(Q) and y=y_(I)+jy_(Q), respectively, Equation (11) becomes:

$\begin{matrix}{{P\left( {\left. r \middle| b_{i} \right. = 1} \right)} = {C{\sum\limits_{d_{I} \in M_{i}^{1}}^{\;}{{\exp\left\lbrack \frac{{y_{I}d_{I}} - \frac{{h}^{2}{d_{I}}^{2}}{2}}{\sigma^{2}} \right\rbrack}{\sum\limits_{d_{Q}}^{\;}{\exp\left\lbrack \frac{{y_{q}d_{q}} - \frac{{h}^{2}{d_{Q}}^{2}}{2}}{\sigma^{2}} \right\rbrack}}}}}} & (12)\end{matrix}$

where the bit to constellation mapping according to (8a) and (8b) havebeen used. The set of real value M_(i) ¹ are the values of each axis(assumed the same for I and Q components) where bit i (b_(i)) takesvalue 1. For example, in 16-QAM. M₁ ¹=M₃ ¹={−1, +1} and M₀ ¹=M₂ ¹={+1,+3}. It is assumed here that the bit of interest affects the in-phasecomponent. The sum over d_(Q) is the sum over all possible values of thequadrature component. A similar expression to Equation (12) can bederived for the probability P(r|b_(i)=0) that r resulted fromtransmission with the i-th bit b_(i) (where iε0, 1, 2 and 3) definingthe transmitted symbol d and being 0.

The likelihood ratio, assuming that bit i affects only the in-phasecomponents, is given by:

$\begin{matrix}{\frac{P\left( {\left. r \middle| b_{i} \right. = 1} \right)}{P\left( {\left. r \middle| b_{i} \right. = 0} \right)} = \frac{\sum\limits_{d_{I} \in M_{i}^{1}}^{\;}{\exp\left\lbrack {\left( {{y_{I}d_{I}} - \frac{{h}^{2}{d_{I}}^{2}}{2}} \right)/\sigma^{2}} \right\rbrack}}{\sum\limits_{d_{I} \in M_{i}^{0}}^{\;}{\exp\left\lbrack {\left( {{y_{I}d_{I}} - \frac{{h}^{2}{d_{I}}^{2}}{2}} \right)/\sigma^{2}} \right\rbrack}}} & (13)\end{matrix}$

b₀ For the most significant bit (MSB) b0 of the 16-QAM constellation,which relates to the sign of the transmitted symbol, M₀ ¹={+1, +3} andM₀ ⁰={−1, −3}, which provides the likelihood ratio for bit b0 as:

$\begin{matrix}{\frac{P\left( {\left. r \middle| b_{0} \right. = 1} \right)}{P\left( {\left. r \middle| b_{0} \right. = 0} \right)} = \frac{{\exp\left\lbrack {\left( {y_{I} - \frac{{h}^{2}}{2}} \right)/\sigma^{2}} \right\rbrack} + {\exp\left\lbrack {\left( {{3y_{I}} - \frac{9{h}^{2}}{2}} \right)/\sigma^{2}} \right\rbrack}}{{\exp\left\lbrack {\left( {{- y_{I}} - \frac{{h}^{2}}{2}} \right)/\sigma^{2}} \right\rbrack} + {\exp\left\lbrack {\left( {{{- 3}y_{I}} - \frac{9{h}^{2}}{2}} \right)/\sigma^{2}} \right\rbrack}}} & (14)\end{matrix}$

where the first and second exponents in the numerator arise from thesymbol constellations d_(I) having real values M=+1 and M=+3,respectively, and the first and second exponents in the denominatorarise from the symbol constellations d_(I) having real values M=−1 andM=−3, respectively.

Taking the logarithmic of Equation (14), the log likelihood ratio λ₀ forb0 is given by:

$\begin{matrix}{\lambda_{0} = {\log \frac{P\left( {\left. r \middle| b_{0} \right. = 1} \right)}{P\left( {\left. r \middle| b_{0} \right. = 0} \right)}}} & (15)\end{matrix}$

Equation (15) can be simplified by approximation to quicken executiontime of the demodulator 223. There are four exponents in Equation (14).Depending on the value of the modified observation y (in particular its1-component y_(I) or its Q-component y_(Q)), only the exponent havinggreater contribution to the probability function in each of thenumerator and the denominator is retained, while the exponent havingless contribution to the probability function in each of the numeratorand the denominator is disregarded. In particular:

-   -   For “small-magnitude” modified observations, that is for        y_(I)<±2, λ₀ can be approximated by disregarding the        contributions from the M=+3 and M=−3 symbol constellations (i.e.        the second exponents in both the numerator and the denominator        in (14)) and relying on the contributions from M=+1 and M=−1        (i.e. the second exponents in both the numerator and the        denominator in (14)) to give:

$\begin{matrix}{{\lambda_{0} \approx {\left\lbrack \frac{\left( {y_{I} - \frac{{h}^{2}}{2}} \right)}{\sigma^{2}} \right\rbrack - \left\lbrack \frac{\left( {{- y_{I}} - \frac{{h}^{2}}{2}} \right)}{\sigma^{2}} \right\rbrack}} = {2{y_{I}/\sigma^{2}}}} & \left( {16a} \right)\end{matrix}$

-   -   For positive and “large-magnitude” modified observations, that        is for y_(I)>+2, λ₀ can be approximated by disregarding the        contributions from M=+1 and M=−3 (i.e. the second and the first        exponents in the numerator and the denominator, respectively in        (14)) and relying on the contributions from M=−1 and M=+3 (i.e.        the first and the second exponents in the numerator and the        denominator, respectively, in (14)) to give:

$\begin{matrix}{{\lambda_{0} \approx {\left\lbrack \frac{\left( {y_{I} - \frac{{h}^{2}}{2}} \right)}{\sigma^{2}} \right\rbrack - \left\lbrack \frac{\left( {{{- 3}y_{I}} - \frac{9{h}^{2}}{2}} \right)}{\sigma^{2}} \right\rbrack}} = {\frac{4y_{I}}{\sigma^{2}} - \frac{4{h}^{2}}{\sigma^{2}}}} & \left( {16b} \right)\end{matrix}$

-   -   For negative and “large-magnitude” modified observations, that        is for y_(I)<−2, λ₀ can be approximated by disregarding the        contributions from M=−1 and M=+3 (i.e. the first and the second        exponents in the numerator and the denominator, respectively, in        (14)) and relying on the contributions from M=+1 and M=−3 (i.e.        the second and the first exponents in the numerator and the        denominator, respectively, in (14))) to give:

$\begin{matrix}{{\lambda_{0} \approx {{- \left\lbrack \frac{\left( {{- y_{I}} - \frac{{h}^{2}}{2}} \right)}{\sigma^{2}} \right\rbrack} - \left\lbrack \frac{\left( {{{- 3}y_{I}} - \frac{9{h}^{2}}{2}} \right)}{\sigma^{2}} \right\rbrack}} = {\frac{4y_{I}}{\sigma^{2}} + \frac{4{h}^{2}}{\sigma^{2}}}} & \left( {16c} \right)\end{matrix}$

-   -   For completeness, for equal power modified observations,        λ₀≈4y_(I)/σ² by disregarding the contribution of M=+/−1 and        relying on the contribution of M=+/−3.

FIGS. 7A and 7B provide a comparison between the optimal LLR (i.e. theLLR given by Equation (14)) and the approximations given by Equations(16a)-(16c). In FIG. 7A, the optimal LLR0 (λ₀) (reference numeral 702)is seen to be closely approximated by the approximated LLR0 (λ₀) givenby Equation (16a) (reference numeral 704) at small magnitudes (i.e.y_(I)±2). Similarly, in FIG. 7B, the optimal LLR0 (λ₀) (referencenumeral 702) is seen to be also closely approximated by the approximatedLLR0 (λ₀) given by Equations (16b) and (16c) (reference numerals 706 aand 706 b) at large magnitudes (i.e. y_(I)>±2). (Note that theapproximated LLR0 (λ₀) at y_(I)=0 is ±4|h|²/σ² and is thusdiscontinuous. In FIG. 7B, the small segment 706 c joining the twosegments 706 a and 706 b is an artefact resulting from the graphingtool)

A generated LLR greater than 0 (i,e. above the upper part of Figure)indicates that the bit b₀ in question is more likely to be a logical 1than a logical 0 (and vice versa). The LLRs are provided to the decoder226 for decoding purposes. The close agreement between the optimal andthe approximated LLRs indicates, in some arrangements, that it may bebeneficial to sacrifice obtaining the optimal values of LLR for the sakeof faster execution of LLR computation according to Equations (16a) to(16c). For completeness, for y_(I) equal to boundary values (e.g.y_(I)=±2), since the small-magnitude and the large-magnitudeapproximations (for small |h|) intersect at y_(I)=±2, as shown in FIG.7A, either approximation is appropriate.

b1

For the next significant bit b1 of the 16-QAM constellation. M₁ ¹={−1,+1} and M₁ ⁰={−3, +3}, which provides the likelihood ratio for bit b1as:

$\begin{matrix}{\frac{P\left( {\left. r \middle| b_{1} \right. = 1} \right)}{P\left( {\left. r \middle| b_{1} \right. = 0} \right)} = {{\exp \left\lbrack {4\frac{{h}^{2}}{\sigma^{2}}} \right\rbrack}\frac{{\exp \left\lbrack {{- y_{I}}/\sigma^{2}} \right\rbrack} + {\exp \left\lbrack {y_{I}/\sigma^{2}} \right\rbrack}}{{\exp \left\lbrack {{- 3}{y_{I}/\sigma^{2}}} \right\rbrack} + {\exp \left\lbrack {3{y_{I}/\sigma^{2}}} \right\rbrack}}}} & (17)\end{matrix}$

where the first and second exponents in the numerator arise from thesymbol constellations d_(I) having real values M=−1 and M=+1,respectively, and the first and second exponents in the denominatorarise from the symbol constellations d_(I) having real values M=−3 andM=+3, respectively.

Taking the logarithmic of Equation (17), the log likelihood ratio λ₁ forb1 is given by:

$\begin{matrix}{\lambda_{1} = {\log \; \frac{P\left( {\left. r \middle| b_{1} \right. = 1} \right)}{P\left( {\left. r \middle| b_{1} \right. = 0} \right)}}} & (18)\end{matrix}$

Like Equation (15), Equation (18) can be simplified by approximation ina similar manner to quicken execution time as follows:

-   -   For positive modified observations, that is for y_(I)>0, λ₁ can        be approximated by disregarding the contributions from the M=−1        and M=−3 symbol constellations (i.e. the first exponents in both        the numerator and the denominator in (17)) and relying on the        contributions from M=+1 and M=+3 (i.e. the second exponents in        both the numerator and the denominator in (17)) to give:

$\begin{matrix}{{\lambda_{1} \approx {\left\lbrack \frac{4{h}^{2}}{\sigma^{2}} \right\rbrack + \left\lbrack {y_{I}/\sigma^{2}} \right\rbrack - \left\lbrack {3{y_{I}/\sigma^{2}}} \right\rbrack}} = {\left\lbrack \frac{4{h}^{2}}{\sigma^{2}} \right\rbrack - \frac{2y_{I}}{\sigma^{2}}}} & \left( {19a} \right)\end{matrix}$

-   -   For negative modified observations, that is for y_(I)<0, λ₁ can        be approximated by disregarding the contributions from M=+1 and        M=+3 (i.e. the second exponents in both the numerator and the        denominator in (17)) to give and relying on the contributions        from M=−1 and M=−3 (i.e. the first exponents in both the        numerator and the denominator in (17)):

$\begin{matrix}{{\lambda_{1} \approx {\left\lbrack \frac{4{h}^{2}}{\sigma^{2}} \right\rbrack + \left\lbrack {{- y_{I}}/\sigma^{2}} \right\rbrack - \left\lbrack {{- 3}{y_{I}/\sigma^{2}}} \right\rbrack}} = {\left\lbrack \frac{4{h}^{2}}{\sigma^{2}} \right\rbrack + \frac{2y_{I}}{\sigma^{2}}}} & \left( {19b} \right)\end{matrix}$

-   -   Combining Equations (19a) and (19b), the LLR for b1 can be        approximated and expressed as:

$\begin{matrix}{\lambda_{1} \approx {\left\lbrack \frac{4{h}^{2}}{\sigma^{2}} \right\rbrack - {\frac{2y_{I}}{\sigma^{2}}}}} & \left( {19c} \right)\end{matrix}$

λ₁(y_(I)=0)=λ₀ ^(large-magnitude)(y_(I)=0)=4|h|²/σ²=4η is termed the“fold point” of the demodulator 223, which is called a foldingdemodulator. That is, the fold point takes the value of the LLR of thenext significant bit λ₁ (or the negative of the most significant bit inlarge-magnitude approximation −λ₀ ^(large-magnitude)) when the in-phasecomponent or quadrature component (depending on which of complementaryLLRs is in question) of the modified observation is zero. The fold pointmay assist in identifying the LLR for the next significant bit.η=|h|²/σ² is the channel power to noise power ratio, or thesignal-to-noise ratio (SNR). The noise power, represented by σ², may beestimated or measured by the noise power estimator 240.

FIG. 7C provides a comparison between the optimal LLR (i.e. Equation(17)) and the approximations given by Equations (19a)-(19c). The optimalLLR1 (λ₁) (reference numeral 708) is seen to be closely approximated bythe approximated LLR1 (λ₁) given by Equation (19c) (reference numeral710) regardless of magnitudes of y_(I).

A LLR greater than 0 (i.e. the upper part of FIG. 7C) indicates that thecorresponding bit b₁ is more likely to be a logical 1 than a logical 0(and vice versa). The LLRs are provided to the decoder 226 for decodingpurposes. The close agreement indicates, in some arrangements, that itmay be beneficial to sacrifice obtaining the optimal values of LLR forfaster execution of LLR computation according to Equations (19a) to(19c).

For completeness, FIG. 7D provides a comparison between the optimal LLR0(λ₀) (reference numeral 702) and the folding LLR0 (i.e. equal powermodified observation) λ₀=4y_(I)/σ² (reference numeral 712). Forillustration purposes, FIG. 7E provides all LLR0 shown in FIGS. 7A-7D onthe same scale. In all of FIGS. 7A-7E, the signal-to-noise ratio is 0dB, such that η=|h|²/σ²=1.

b2 and b3

Similarly, a skilled person in the art would appreciate that, theexpressions λ₂ and λ₃ for the LLRs for bit b2 and b3 are similar tothose of λ₀ and λ₁, respectively, except that y_(I) is replaced byy_(Q). λ₂ and λ₀ are complementary LLRs, and are the most significantbits. Similarly, λ₃ and λ₁ are also complementary LLRs, and are the nextsignificant bits (and also the least significant bits in the case of16-QAM)).

FIG. 6B illustrates a logic block diagram of a subsystem 600 includingan observation modifier 222 and related components in facilitatingcalculation of Equations (16a)-(16c) and Equations (19a)-(19c). In thefirst branch 602, the observation modifier 222 obtains a receivedobservation r. The observation modifier 222 multiplies the receivedobservation r by the conjugate of the channel estimate h obtained fromthe second branch 604 from, for example, a channel estimator (not shown)to generate a modified observation y. Either the I or Q component of y,depending on which of the complementary LLRs is to be calculated, issaturated by a saturate block 608. In the second branch 604, the channelestimate is conjugated by a conjugate block 610 and provided to thefirst branch 602 for the multiplication operation described above. Stillin the second branch 604, the channel estimate is multiplied by itsconjugate before being saturated by a saturate block 612 to provide thechannel power (represented by |h|²). In the third branch 606, a measuredor estimated noise power (represented by σ²) is obtained, for examplefrom a noise power estimator, and is inverted by an inverse block 618.The inverse block 618 may be a look up table (LUT). The inverted noisepower (1/σ²) is provided to the second branch 604 for multiplying withthe channel power to estimate the signal-to-noise ratio (represented by|h|²/σ²). The inverted noise power (1/σ²) is provided to the firstbranch 602 to calculate y_(I)/σ² or y_(Q)/σ². The outputs of the firstbranch 602 (i.e. y_(I)/σ² or y_(Q)/σ² and the second branch (i.e.|h|²/σ²) facilitate the calculation of LLR approximations given byEquations (16a)-(16c) and also (19a)-(19c).

From the foregoing, it is apparent that the LLRs generated based on themodified observation v has reduced dependence on the magnitude of thechannel h.

64 QAM

The foregoing examples are directed to a 16-QAM system. It should beapparent to a skilled person to derive, for a 64-QAM systems, similarexpressions for λ₀, λ₁, λ₂, λ₃, λ₄ λ₅.

Further Simplification

Equation (19c) indicates that λ₁ (and hence λ₃) is given by oneapproximation. On the other hand, Equations (16a)-(16c) indicate that λ₀(and hence λ₂) can be given by three different approximations, dependingon the magnitude of the real part (and the imaginary part) of themodified observation y. It would further simplify the demodulator 223where λ₀ (and hence λ₂) is approximated by only one of theapproximations. FIG. 8 illustrates a comparison between the packet errorrate (PER) 802 where λ₀ (and hence λ₂) is approximated by all 3approximations depending on the magnitude of the real part (or theimaginary part) of the modified observation, and the PER 804 where λ₀(and hence λ₂) is approximated by Equation (16a), that is, thesmall-magnitude approximation, regardless of on the magnitude of thereal part (or the imaginary part) of the modified observation y. λ₁ andλ₃ are approximated by Equation (19c). As illustrated in FIG. 8, thereis only a small degradation in PER where only the small-magnitudeapproximation of λ₀ and λ₂ is used. For example, at receiver power of−89 dBm, the difference is between about 0.014 (for PER 802) and 0.015(for PER 804).

Spatial Diversity

For spatially diversified receivers having multiple antennas, such as afirst antenna 221 and a second antenna 221′, each antenna generallyperceives a different channel h and is associated with a differentantenna noise power n. The equations above may be extended withmodifications as follows and the demodulation may be adaptedaccordingly. For the α-th antenna, the received observation r of symbold becomes:

r[i,j,α]=h[i,j,α]d[i,j]+n[i,j,α]  (20)

using like designations of h, d, i and j as Equation (1).

The modified observation p for calculating an effective LLR across allantennas can be given by a weighted sum of individual modifiedobservation y for each antenna α in the form of Equation (9). Inparticular, the weights are given by the antenna noise power representedby σ_(α) ²:

ρ[i,j]=Σ _(α) y[i,j,α]/σ _(α) ²=Σ_(α) h*[i,j,α]r[i,j,α]/σ _(α) ²  (21)

In other words, in a spatially diversified receiver, the observationmodifier 222 may generate an effective modified observation p as aweighted sum, based on the product of the observation received byindividual antennas and the complex conjugate of the channel estimate ofthe channel perceived by the respective antenna, normalised by therespective antenna noise power.

The noise power estimator may be configured to estimate or measure theeffective SNR calculated as the sum of SNRs of individual antennas:

η[i,j]=Σ _(α) h*[i,j,α]h[i,j,α]/σ _(α) ²=Σ_(α) |h[i,j,α] ²/σ_(α) ²  (22)

For a 16-QAM system, the LLR module 225 may be configured to calculatethe complementary LLRs for the most significant bits (i.e. λ₀ and λ₂)as:

λ₀ [i,j]≈2ρ_(I) [i,j]  (23a)

λ₂ [i,j]≈2ρ_(Q) [i,j]  (23b)

where ρ_(I) and ρ_(Q) are the real and imaginary parts of the modifiedobservation ρ. It is noted that Equation (23a) (and hence similarly(23b)) takes the same form as Equation (16a) except that the effectivemodified observation is based the noise-weighted sum given by Equation(21), rather than Equation (9). Further, the LLR module 225 may beconfigured to calculate the complementary LLRs for the least significantbits (i.e. λ₁ and λ₃) as:

λ₁ [i,j]≈4η[i,j]−|λ ₀ [i,j]|  (23c)

λ₃ [i,j]≈4π[i,j]−|λ ₂ [i,j]|  (23d)

It is noted that Equation (23a) (and hence similarly (23b)) takes thesame form as Equation (19a) except that the effective modifiedobservation is based on the weighted sum given by Equation (21), ratherthan Equation (9), and also that the SNR η is the sum of channel powerto noise power ratio, summed over all antennas.

For a 64-QAM system, it should be apparent to a skilled person that theLLR module 225 may be configured to calculate the complementary LLRs forthe most significant bits (MSB), the next significant bits (NSB) and theleast significant bits (LSB) following a similar and slightly modifiedprocedure above for 16-QAM:

For MSB: λ₀ [i,j]≈2ρ_(I) /[i,j]  (24a)

λ₃ [i,j]≈2ρ_(Q) [i,j]  (24b)

For NSB: λ₁ [i,j]≈4η[i,j]−|λ ₀ [i,j]|  (24c)

λ₄ [i,j]≈4η[i,j]−|λ ₃ [i,j]|  (24d)

For LSB: λ₂ [i,j]≈2η[i,j]−|λ ₁ [i,j]|  (24e)

λ₅ [i,j]≈2η[i,j]−|λ ₄ |[i,j]|  (24f)

Again, Equations (24a) to (24d) have the same form as Equations (23a) to(23d). Equations (24e) and (24f) follow similar approximations. FIG. 9illustrates a logic block diagram for an example of a demodulator 900including an observation modifier 902 and a LLR module 904 for amultiple-antenna implementation in a 64-QAM system. The observationmodifier 902 is configured to generate a modified observation accordingto Equation (21). The LLR module 904 is configured to generate LLRsaccording to Equations (22)-(24).

The observation modifier 902 includes a first subsystem 902 a for afirst antenna A and a second subsystem 902 b for a second antenna B.Each of the subsystems 902 a and 902 b is similar to the subsystem 600FIG. 6B (without showing the saturate blocks, the inverse LUT block, theshift block or Calc exponent block for simplification), except thattheir outputs ρ_(A) and ρ_(B) are noise-weighted modified observations.In other words, ρ_(A) and ρ_(B) are individual terms for the respectiveantennas in the summation given by Equation (21). The observationmodifier 902 computes the summation and provides an effective modifiedobservation p as an output. The first subsystem 902 a and the secondsubsystem 902 b, like the subsystem 600, also provides estimates of thesignal-to-noise ratio for each antenna (represented by |h_(A)|²/σ_(A) ²and |h_(B)|²/σ²).

As shown in FIG. 9, the LLR module 904 obtains and add |h_(A)|²/σ_(A) ²and |h_(B)|²/σ_(B) ² to generate η according to Equation (22). The LLRmodule 904 also obtains the real and imaginary parts of the effectivemodified observation p. According to Equations 24(a)-(e), the LLR module904 calculates LLR0, LLR2 and LLR4 (i.e. LLRs for the MSB, NSB and LSB)based on the real part and calculates LLR1, LLR3 and LLR5 (i.e. LLRs forthe MSB, NSB and LSB) based on the imaginary part:

-   -   In the first and the fourth branches 904 a and 904 d, the LLR0        and LLR1 are calculated based on Equations (24a) and (24b). For        simplification, a multiplication block by a factor of 2        (according to Equations 24(a) and (b)) is not shown for LLR0 and        LLR. LLR0 and LLR1 therefore is shown to take the real and        imaginary parts, respectively, of the effective modified        observation ρ directly.    -   In the second and the fifth branches 904 b and 904 e, the LLR2        and LLR3 are calculated based on Equations (24c) and (24d). For        simplification, a multiplication block by a factor of 2        (according to Equations 24(a) and (b)) to calculate LLR0 and        LLR1 is not shown. LLR2 and LLR3 are obtained by providing LLR0        and LLR1 to an absolute value block (906 a and 906 b) and adding        to a scaled η generated by the LLR module 904. The scale is 4        according to Equations (24c) and (24d).    -   In the third and the sixth branches 904 c and 904 f, the LLR4        and LLR5 are calculated based on Equations (24e) and (24f). LLR4        and LLR5 are obtained by providing LLR2 and LLR3 to an absolute        value block (906 c and 906 d) and adding to a scaled η generated        by the LLR module 904. The scale is 2 according to Equations        (24e) and (24f).

One or more of the components of the receiver 220 or the demodulator 900may be implemented as software, such as a computer program includinginstructions stored in a non-transitory computer-readable medium andexecutable by the one or more processors. In one example, thenon-transitory computer-readable medium is a memory or storage module,such as volatile memory including a random access memory (RAM),non-volatile memory including read-only memory (ROM), or a harddisk. Theone or more processors may be one or more computer processing units(CPUs). Alternatively or additionally the one or more of the componentsof the receiver 220 may be implemented as hardware, such as using one ormore digital signal processors (DSPs), application-specific integratedcircuits (ASICs) or field-programmable gate arrays (FPGAs).

What is claimed is:
 1. A method comprising: receiving, at a firstantenna, an observation of a symbol transmitted across a wirelesscommunications channel perceived by the first antenna; generating amodified observation based on a product of the received observation andthe complex conjugate of a channel estimate of the channel; andgenerating, based on the modified observation and the channel estimate,log-likelihood ratios (LLRs) for a maximum-likelihood-based decoder todecode.
 2. The method of claim 1 wherein generating log-likelihoodratios includes generating a LLR associated with a most significant bitof the symbol.
 3. The method of claim 2 wherein the log-likelihood ratio(LLR) associated with the most significant bit is generated based on theratio of the real part (y_(I)) or imaginary part (y_(Q)) of the modifiedobservation to Gaussian-distributed noise power of the channel.
 4. Themethod of claim 3 wherein the LLR associated with the most significantbit is estimated to be 2 y_(I)/σ² or 2 y_(Q)/σ², where σ² represents theGaussian-distributed noise power of the channel.
 5. The method of claim3 wherein the LLR associated with the most significant bit is estimatedto be 4 y_(I)/σ² or 4 y_(Q)/σ², where σ² represents theGaussian-distributed noise power of the channel.
 6. The method of claim3 wherein the LLR associated with the most significant bit is estimatedto be 4 y_(I)/σ²−4|h|²/σ² or 4 y_(Q)/σ²−4|h|²/σ², where σ² representsthe Gaussian-distributed noise power of the channel and |h| representsthe magnitude of the channel estimate.
 7. The method of claim 3 whereinthe LLR associated with the most significant bit is 4 y_(I)/σ²+4|h|²/σ²,where σ² represents the variance of the Gaussian-distributed noise powerof the channel and |h| represents the magnitude of the channel estimate.8. The method of claim 1 wherein generating log-likelihood ratiosincludes generating a LLR associated a next most significant bit of thesymbol.
 9. The method of claim 8 wherein the LLR associated with thenext most significant bit is generated based on the ratio of the realpart (y_(I)) or imaginary part (y_(Q)) of the modified observation toGaussian-distributed noise power of the channel.
 10. The method of claim9 wherein the LLR associated with the most significant bit is 4|h|/σ²−2y_(I)/σ² or 4|h|/σ²−2 y_(Q)/σ², where σ² represents the variance of theGaussian-distributed noise power of the channel and |h| represents themagnitude of the channel estimate.
 11. The method of claim 3 wherein theGaussian-distributed noise power is estimated or measured.
 12. Themethod of claim 1 further comprising receiving, at a second antennalocated separately from the first antenna, another observation of thesymbol transmitted across a wireless communications channel perceived bythe second antenna, and wherein generating a modified observationincludes generating the modified observation also based on a product ofthe other received observation and the complex conjugate of a channelestimate of the channel perceived by the second antenna.
 13. The methodof claim 12 wherein generating the modified observation is based on aweighted sum of (a) the product of the received observation and thecomplex conjugate of a channel estimate of the channel and (b) theproduct of the other received observation and the complex conjugate of achannel estimate of the channel perceived by the second antenna.
 14. Themethod of claim 13 wherein the weighted sum is based on weightsassociated with the noise power as observed by the respective antennas.15. The method of claim 1 further comprising decoding, by amaximum-likelihood-based decoder, the LLRs.
 16. A wireless receivercomprising: a first antenna for receiving an observation including asymbol transmitted across a wireless communications channel perceived bythe first antenna; an observation modifier for generating a modifiedobservation based on a product of the received observation and thecomplex conjugate of a channel estimate of the channel; a log-likelihoodratio (LLR) module generating log-likelihood ratios (LLRs) based on themodified observation; and a maximum-likelihood-based decoder fordecoding the symbol based on the LLRs.
 17. The wireless receiver ofclaim 16 further comprising a second antenna for receiving anotherobservation of the symbol transmitted across a wireless communicationschannel by the second antenna, the first antenna and the second antennaare located to provide spatial diversity, wherein the observationmodifier is configured to generate the modified observation based onweighted sum of (a) the product of the received observation and thecomplex conjugate of a channel estimate of the channel and (b) theproduct of the other received observation and the complex conjugate of achannel estimate of the channel perceived by the second antenna.
 18. Thewireless receiver of claim 17 wherein the weighted sum is based onweights associated with the noise power as observed by the respectiveantennas.
 19. The wireless receiver of claim 16 further comprising amaximum-likelihood-based decoder for decoding the LLR.
 20. Anon-transitory machine-readable medium comprising instructionsexecutable by one or more processors, the instructions including thesteps of: receiving, at a first antenna, an observation of a symboltransmitted across a wireless communications channel perceived by thefirst antenna; generating a modified observation based on a product ofthe received observation and the complex conjugate of a channel estimateof the channel; and generating, based on the modified observation andthe channel estimate, log-likelihood ratios (LLRs) for amaximum-likelihood-based decoder to decode.